A body with no sea level

Every height on Earth is measured from a surface that nobody had to choose. Water finds the equipotential surface by itself; mean sea level is a physical object; the geoid is a description of something that already exists, and the reference ellipsoid was then defined to be a level surface so that the geometrical datum and the physical one would agree.

That arrangement is so convenient that it is easy to mistake for the natural order of things. It is not available anywhere else. Mars has no sea, so the zero of Martian elevation is a decision, and it has been made twice this century: an atmospheric-pressure surface — the level at which the mean pressure equals the triple point of water — until 2001, and an equipotential surface tied to the mean equatorial radius since. Those are different surfaces. The elevation of a Martian place changed by kilometres without the ground moving.

What this site can compute, rather than cite, is the relation between a body’s shape and its gravity field, and how far each body is from having them agree.

Clairaut’s theorem as a prediction rather than a check #

The site already carries Clairaut’s theorem in its terrestrial form, where it relates the geometrical flattening to the gravity flattening and is used to audit the normal field: the ellipsoid is a level surface shows the two summing to 5m/2 with a residual that is second order in f, which is what a first-order theorem is entitled to be wrong by.

Turned round, the same theorem is a prediction:

$$f \approx \tfrac{3}{2}J_2 + \tfrac{m}{2}, \qquad m = \frac{\omega^2 a^3}{GM}$$

with J₂ the dynamical form factor and m the ratio of centrifugal to gravitational acceleration at the equator. Both inputs are properties of the field, and the output is a property of the shape. For a body in hydrostatic equilibrium — a fluid figure, settled — the prediction is the shape.

The Earth’s numbers: J₂ = 1.0826 × 10⁻³ and m = 3.4614 × 10⁻³, giving a predicted flattening of 1/298.10 against a published 1/298.26. The disagreement is 0.055 per cent, which is twelve metres at the pole.

Twelve metres is small enough to be second-order theory and nothing else. The Earth is a fluid figure to the accuracy a first-order theorem can see.

Mars is not #

Mars’s numbers: J₂ = 1.9554 × 10⁻³, m = 4.5953 × 10⁻³, predicted flattening 1/191.2. Its published radii give 1/169.9. The prediction is out by 12.5 per cent, which at the pole is 2.23 kilometres.

That is not a failure of the theorem and it is not a bad measurement. It is the finding: Mars is not in hydrostatic equilibrium, because it carries Tharsis — a volcanic province the size of a continent and several kilometres high, with a corresponding mass excess that the rest of the body has not relaxed away. A body carrying a load like that has a figure its own spin cannot explain, and its reference ellipsoid is therefore not one of its level surfaces.

Every consequence a coordinate system has follows from that one sentence.

A second route to the same conclusion #

The site’s normal-field machinery takes four constants — a, f, GM and ω — and derives everything else in closed form, including J₂ itself. It was written for the Earth and nothing in it is about the Earth, which is a claim worth testing by handing it a planet.

Given Mars’s radii and its GM and rotation rate, the machinery returns J₂ = 2.392 × 10⁻³. The observed J₂ is 1.955 × 10⁻³. The derivation is 22 per cent high, and the reason is the same one: it computes the field of the level ellipsoid that has Mars’s shape, and Mars’s shape is not a level surface, so the field it would have if it were is not the field it has.

Two independent routes, one from shape to field and one from field to shape, disagreeing by the same fraction in opposite directions. That is the site’s usual standard of evidence, applied to a claim about a planet rather than about a projection.

The reason to trust the machinery when it is pointed at Mars is that it can be checked where checking is possible. On the Earth the same four constants reproduce the published equatorial and polar gravity, the published J₂ and the published polar radius to ten significant figures, none of which are inputs.

The distinction matters because a reader is entitled to ask what a 12.5 per cent disagreement on Mars is evidence of. It is not evidence about the arithmetic, which is exact and audited; it is not evidence about the constants, which are published to more digits than the comparison uses; and it is not evidence about the theorem, which the Earth’s case shows holding to a part in two thousand at a comparable flattening. What is left is the body.

What a zero of height is, once it has to be chosen #

Three candidate surfaces are available on any body, and on Earth they nearly coincide, which is why the choice has never had to be argued in public.

A sphere of stated radius. The Moon’s coordinates use one: 1,737.4 km, by convention. It is simple, it is exactly reproducible, and it is a surface no physical process would ever pick — the Moon’s own J₂ implies a level flattening of 1/3,241, which puts the level surface 536 metres below the sphere at the pole.

An ellipsoid fitted to the topography. This is the Martian ellipsoid, and it is what most planetary maps are drawn on. It is a least-squares fit to a shape model, which makes it a summary of the topography rather than a physical surface, and the 2.23 km gap above is the amount by which those two ideas differ on Mars.

An equipotential surface. The physically meaningful one: the surface water would follow, if there were water. It needs a gravity field model, which is a data product exactly as a geoid model is, and it needs one further decision that the Earth’s case hides — which equipotential. On Earth the sea picks the value of the potential; on Mars a number has to be chosen, and the choice made in 2001 was the potential of the surface whose mean equatorial radius is 3,396.19 km.

What was computed, and how #

Two constants per body are citations and everything else is derived. GM, ω and J₂ are taken as published, in the same sense the IAU radii already are: astronomy-celestial-mechanics.com owns these bodies as observed objects, and nothing here computes an orbit, a mass or a rotation rate. What is computed is what those numbers imply for the surface a coordinate is measured from, which is this site’s question.

m is computed rather than quoted, as ω²a³/GM, and it is worth looking at on its own. On Earth the spin contributes 51.6 per cent of the predicted flattening and the mass arrangement 48.4 per cent; on Mars the spin contributes 43.9 per cent. The two mechanisms are comparable in size on both bodies, which is why a theory that dropped either would be useless.

The predicted flattening is a first-order expression and the comparison respects that. The theorem’s next term is of order f², which for Mars is 3 × 10⁻⁵ against a flattening of 5.9 × 10⁻³ — half a per cent of the answer, against a disagreement of 12.5 per cent. The Earth’s residual, measured in the audit above at 2.74 f², is the empirical version of that bound and is what makes the Martian excess twenty times too large to be truncation.

The gap is reported at the pole because that is where a difference in flattening is largest and because it is the number a mapping decision actually feels. At the equator the two surfaces agree by construction: both are fitted to the same equatorial radius.

The refusal is Venus. Its measured flattening is zero to the precision of the published radii, and its predicted one is 6.7 × 10⁻⁶ — 41 metres on a body 6,000 km across, which is a tenth of the thickness of the layer of rock a radar altimeter is even sensitive to. A test that reported a large disagreement there would be reporting its own noise, and the machinery’s not doing so is what licenses the Martian number.

Three kinds of disagreement, and only one of them is about the body #

Running the prediction across every body the site carries a gravity field for gives five rows, and reading them as a single ranking of “how far from equilibrium” would be wrong in three different ways.

Jupiter’s is the theorem’s. Its flattening is a fifteenth, the theorem is a series in the flattening, and 2.6 per cent is about what a dropped second-order term is worth at that size. Nothing in that row is evidence about Jupiter.

Mars’s is the body’s, and it is the only row where that is true. Its flattening is small enough that the truncation is worth half a per cent, so a 12.5 per cent excess is twenty times too large to be the theory, and the Earth’s row is the control that says so at a comparable flattening.

The Moon’s and Venus’s are the datum’s. Their published flattening is not a small number that disagrees with the prediction; it is exactly zero, because both bodies’ coordinates are referred to a sphere by convention. So the gap in those two rows is not a statement about how far the body is from equilibrium at all — it is the size of the convention, which is precisely what this essay set out to measure. A lunar coordinate is published on a surface that the Moon’s own gravity field puts 536 metres above the pole of the level figure, and nobody thinks the Moon is a sphere.

Those two rows also isolate the other half of Clairaut’s expression. On the Earth the spin contributes 51.6 per cent of the predicted flattening and the mass arrangement 48.4; on Jupiter the spin contributes two thirds. On the Moon it contributes 1.2 per cent and on Venus 0.5, because both turn slowly enough that the centrifugal term nearly vanishes — Venus’s day is longer than its year. Their predicted figures are therefore almost pure J₂: a shape implied by how the mass is arranged and not at all by how the body spins. For the Moon that prediction is known to be a fossil rather than a current equilibrium, which is one more reason its sphere has never been replaced.

Jupiter, and where the first-order theory stops #

Jupiter’s flattening is 1/15.4, which is not small, and Clairaut’s first-order theorem is a series in exactly that quantity. The prediction comes out 2.6 per cent from the published figure — 126 kilometres — and almost all of that is the theorem’s truncation rather than the planet’s departure from equilibrium. Jupiter is a fluid figure; it is simply too flattened for one term.

That is worth stating rather than burying, because it is the boundary of the argument. The Martian number is evidence about Mars only because Mars’s flattening is small enough for the first-order theory to be trusted to a per cent, which the Earth’s case demonstrates directly. On a body flattened by a tenth the same comparison would prove nothing.

The same boundary shows up in a second, unrelated measurement, which is the reason to trust it as a boundary rather than as an artefact of this calculation. The error of applying spherical formulae to a body’s own latitudes follows a law of 2f radians, and it holds to a third of a per cent for Mercury, the Earth and Mars while departing by 3.3 and 5.1 per cent for Jupiter and Saturn. The same two bodies leave the regime each time, because a first-order statement is a first-order statement wherever it appears.

Where the model stops #

No geoid, and no areoid. The gap computed here is between the published ellipsoid and the level ellipsoid of the same equatorial radius, both of which are analytic surfaces. The actual Martian equipotential surface departs from either by kilometres in places, and computing that needs a spherical-harmonic gravity model, which is a data product with a truncation degree in it — the citation this site declined in height above what and declines again here. What is computed is the size of the discrepancy that makes the choice necessary, not the shape of the surface eventually chosen.

The topography is not modelled. Saying that the excess flattening is Tharsis is an attribution, not a measurement: this site has not integrated the mass of Tharsis and shown that it accounts for 2.23 kilometres of polar radius. That calculation is done in the planetary geodesy literature and its inputs are a shape model and a gravity model, neither of which is available here on the site’s own terms.

Two of the five rows cannot be read as measurements of a body. The Moon’s and Venus’s published flattenings are zero because their coordinates are referred to a sphere, not because either body has been measured as spherical, and this site carries no independent shape model for either. So the gap quoted for them is the distance between a convention and a prediction, with the third quantity — the actual figure — absent. The Moon in particular is known from laser ranging to carry a figure its present rotation cannot explain, and nothing here measures it; the row says only how far the adopted sphere is from the surface the Moon’s own field would settle into, which is the question the essay asks and not the question of what shape the Moon is.

The polar gap is a difference between two analytic surfaces of the same equatorial radius, so it is zero at the equator by construction and largest at the pole, and it is not the error in anybody’s elevation. An elevation’s error depends on which surface the elevation was actually referred to, and on Mars that has changed once in the period this site’s sources cover.

A triaxial body has no version of this at all. Vesta and Phobos are far too small for hydrostatic equilibrium to be a useful idea and far too irregular for one equipotential to be a sensible datum. Their published coordinates use a fitted triaxial figure, and the previous rung is where the machinery for measuring on one lives.

Who found it, and when #

Clairaut published Théorie de la figure de la Terre in 1743, and its central theorem was the first result to connect a measurement anybody could make — the variation of gravity with latitude — to a quantity nobody could see directly, the flattening of the Earth. It settled an argument that had run for fifty years between the Cassinis, who measured the Earth as prolate, and the Newtonians, who predicted it oblate.

Its planetary use is recent, and Mars is the case that forced it. Mariner 9 in 1971 showed the Tharsis rise; the gravity field that came with it showed a J₂ that did not match the figure, and the mismatch has been the standard evidence for a non-hydrostatic Mars ever since. The Mars Orbiter Laser Altimeter, from 1997, produced the topography that made the modern areoid definable at all, and the current convention dates from a working-group decision in 2001.

The Moon’s sphere is older and more pragmatic: it was fixed by convention before there was any way to do better, and it has been kept because changing a datum invalidates every published coordinate, which is the argument a published coordinate is a result makes about the terrestrial case.

Where the ladder goes next #

This rung and the one before it exhaust what the site set out to say about bodies: what a coordinate on one means, and what the surface it is measured from is. Both essays lean on a quantity that has appeared in every field of this site and has never been examined as a variable — the curvature of the surface being mapped, which on a sphere is one number, on an ellipsoid a function of latitude, and on Vesta a function of two coordinates varying by a factor of nearly three.

The impossibility argument this site is built on was made on a sphere, where the curvature is constant. Whether it survives a surface where it is not is the next question, and the answer changes what how small is flat enough means.